An energy functional for Lagrangian tori in $$\mathbb {C}P^2$$

A two-dimensional periodic Schrödingier operator is associated with every Lagrangian torus in the complex projective plane \({\mathbb C}P^2\). Using this operator, we introduce an energy functional on the set of Lagrangian tori. It turns out this energy functional coincides with the Willmore functional \(W^{-}\) introduced by Montiel and Urbano. We study the energy functional on a family of Hamiltonian-minimal Lagrangian tori and support the Montiel–Urbano conjecture that the minimum of the functional is achieved by the Clifford torus. We also study deformations of minimal Lagrangian tori and show that if a deformation preserves the conformal type of the torus, then it also preserves the area, i.e., preserves the value of the energy functional. In particular, the deformations generated by Novikov–Veselov equations preserve the area of minimal Lagrangian tori.